Trigonometry

posted by Joe

Let N be a 5-digit palindrome. The probability that N is divisible by 4 can be expressed as \frac{a}{b}, where a and b are coprime positive integers. What is the value of a+b?

1. MathMate

There are 900 5-digit palindromes, from 10001, 10101 , 10201, 10301, ... to 99899, 99999.

Out of these 900 palindromes, divisibility is determined by the last four digits, namely 00, 01, 02, ...99.
The probability of divisibility by 4 for digits of 00 to 99 is 1/4.

So a=1, b=4. What is a+b?

2. Joe

I calculated it and a got a totally different answer

3. MathMate

Did you use the sine law?

4. Joe

I think I did use the sine law but I am just not sure how to get a+b?

5. MathMate

What is the probability, i.e. a and b?

6. MathMate

I am curious how you find the probability of divisibility of palindromes by 4 using the sine law. Can you kindly show your work?

7. stranger

Suppose N = xyzyx, where x is nonzero but y and z could be any digit.
Then there are 9(10^2) = 900 possible palindromes to consider.

Recall that a number is divisible by 4 iff its last two digits are divisible by 4. Thus, if N is divisible by 4, then "yx" must be of the form "04", "08", "12", "16", ..., "92", "96", except it cannot end in a 0 (so we must omit "20", "40", "60", and "80"). Hence, there are (96/4 - 4) = 20 possible values of "yx". Since z is unrestricted, we multiply by 10 to yield 200 palindromes that satisfy this criteria.

Thus, P(N is divisible by 4) = 200/900 = 2/9.
Since a = 2 and b = 9, we have a+b = 11.

Similar Questions

1. Calulus

Given \displaystyle \int_0^{\frac{3\pi}{2}} x^2\cos x \, dx = a - \frac{b\pi^2}{c}, where a, b and c are positive integers and b and c are coprime, what is the value of a + b + c?
2. Geometry

Six standard six-sided die are rolled. Let p be the probability that the dice can be arranged in a row such that for 1\leq k \leq 6 the sum of the first k dice is not a multiple of 3. Then p can be expressed as \frac{a}{b} where a …
3. Algebra

Joe picks 2 distinct numbers from the set of the first 14 positive integers S = \{1,2,3,\ldots,14\}. The probability that the sum of the 2 numbers is divisible by 3 can be expressed as \frac{a}{b}, where a and b are coprime positive …
4. Calculus

Given f(x) = \frac{x^3-2x+5}{x+4} and f’(3) = \frac{a}{b}, where a and b are coprime positive integers, what is the value of a+b?
5. Geometry

Two points are chosen uniformly at random on the unit circle and joined to make a chord C_1. This process is repeated 3 more times to get chords C_2, C_3, C_4. The probability that no pair of chords intersect can be expressed as \frac{a}{b} …
6. Geometry

Two points are chosen uniformly at random on the unit circle and joined to make a chord C_1. This process is repeated 3 more times to get chords C_2, C_3, C_4. The probability that no pair of chords intersect can be expressed as \frac{a}{b} …
7. Trigonometry

ABC is a right triangle with AB = 2, BC = 3\sqrt{5}, AC = 7 and \angle ABC = 90^\circ. If \cos \angle BAC = \frac{a}{b}, where a and b are coprime positive integers, what is the value of a + b?
8. Maths

Two players each flip a fair coin. The probability that they get the same result can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?
9. Geometry

Two players each flip a fair coin. The probability that they get the same result can be expressed as a b where a and b are coprime positive integers. What is the value of a+b ?